Base ten blocks

Type a number up to 9,999, like `2345`, and it's drawn as base ten blocks: big cubes for thousands, flats for hundreds, rods for tens and small cubes for ones. Type a sum, like `47 + 38` or `52 - 17`, and you see both numbers, the answer, and every trade in between, written out place by place.

The printable set below has two big cubes, four flats, twenty rods and forty unit cubes to cut out, for a class without enough plastic ones. Press Print chart.

Each block is ten of the one before it. Ten unit cubes in a line make a rod, worth 10. Ten rods side by side make a flat, worth 100. Ten flats stacked make a big cube, worth 1,000. So 345 is three flats, four rods and five units, and the digits of the number are just how many of each block there are.

That's the point of them. A child can see that the 3 in 345 means three hundred because it's three flats, each obviously bigger than a rod. The place value chart shows the same idea in columns.

Adding with blocks means putting both piles together. In `47 + 38`, the ones come to fifteen unit cubes. That's too many for one place, so ten of them are traded for a rod, leaving five. The tens are then four rods plus three plus the one traded in: eight. The answer is 85.

Subtracting runs the other way. In `52 - 17` there are only two units, not enough to take seven, so a rod is broken into ten units, making twelve. Twelve take seven is five; four rods take one is three. Written on paper, those trades are the carries and borrows of addition with regrouping and subtraction with regrouping.

Catherine Stern first described base ten blocks in 1949, though Maria Montessori had already used golden beads strung into the same shapes. They're often called Dienes blocks after Zoltán Dienes, who popularised them from the 1950s. Dienes wanted blocks in other bases too, two, three and so on, so that children would see ten as one choice among many. Those multibase blocks had their moment in the New Math of the 1960s. The base ten set is the one that stayed.